Vanek cashed in on Buffalo's seventh power-play opportunity with 3:53 to go after defenseman James Vandermeer was sent off for cross-checking him behind the Phoenix net.
Scottie Upshall received four minutes in penalties in the final period, wasting a great effort from LaBarbera while leaving new Coyotes coach Dave Tippett livid.
"We're in a game that our goaltender is giving us a chance to win and we go out there and take six more minutes in penalties? That's disgusting," Tippett said. "That's giving games away. That's bad discipline on our part and that's situations that can't happen."
LaBarbera made his first start for the Coyotes after signing as a free agent. He finished with 37 saves, many coming in spectacular fashion.
"Anytime you join a new team you want to show them a good effort," he said. "It would have been nice to win, but it was one of those games where I needed one more save."
Clarke MacArthur added a goal and assist for the Sabres, who rebounded from a season-opening overtime loss to Montreal.
Matthew Lombardi scored for the Coyotes.
Phoenix (2-1), embroiled in off-ice controversy since filing for bankruptcy in May and nearly winding up in nearby Hamilton, Ontario, then enduring a coaching change just before the season, lost for the first time and was denied its best start since 2003. The Coyotes return home on Saturday to play their first regular season game in Glendale since entering bankruptcy protection.
"We've been told this first game's already sold out," captain Shane Doan said. "We're excited about that. It's obviously a difficult situation, and nobody wants to be in it."
Vanek scored from the slot, one-timing MacArthur's pass from underneath the goal line.
"We had chances but just couldn't get them," Vanek said. "We stuck to our systems, stuck to our game and got rewarded in the end."
Phoenix forward Petr Prucha was knocked out of the game 32 seconds into the second period after taking a hit from Sabres forward Patrick Kaleta. Martin Hanzal jumped Kaleta and received 19 minutes worth of penalties, leaving Phoenix two players short for most of the period while giving the Sabres a 4-minute power play.
Buffalo peppered LaBarbera with six shots during that stretch, but came up empty.
"We could have had three or four in the second period, but their goalie was unbelievable," MacArthur said.
Tippett didn't have an update on Prucha, who was leveled after playing the puck along the side boards before turning to skate up ice.
"If it was a high hit to the head, the league will look after it," Tippett said.
After two scoreless periods the Coyotes broke through on Lombardi's power-play goal 2:37 into the third period.
MacArthur tied it 97 seconds later after Sabres rookie defenseman Tyler Myers was able to keep the puck in the Phoenix zone before feeding MacArthur. Myers earned his first career NHL assist on the play, but was shaken up after the pass after taking a hit by Doan. Myers was on the ice for a couple of minutes then left under his own power.
Miller kept the game scoreless in the second period with 4:42 left, robbing Radim Vrbata with a glove save while on his stomach.
NOTES: Referee Bill McCreary officiated in his 1,600th NHL game. ... Buffalo C Adam Mair missed his second game because of a hip injury, but is close to returning. ... The Sabres were also missing D Toni Lydman (hand). ... Buffalo won both games last season by a 7-1 margin. ... Miller had not allowed a goal to the Coyotes in his seven-year career.
|PPG - Matthew Lombardi (2) ASST: Adrian Aucoin (2), Shane Doan (2)|
1 - 0 PHX
|Clarke MacArthur (1) ASST: Tyler Myers (1)|
1 - 1 Tie
|PPG - Thomas Vanek (1) ASST: Clarke MacArthur (1), Derek Roy (2)|
2 - 1 BUF
|Patrick Kaleta Fighting against Martin Hanzal|
|Martin Hanzal Fighting against Patrick Kaleta|
|Martin Hanzal Instigator against Patrick Kaleta|
|Martin Hanzal Instigator - face shield against Patrick Kaleta|
|Martin Hanzal Misconduct|
|Steve Montador Hooking against Mikkel Boedker|
|Henrik Tallinder Tripping against Scottie Upshall|
|Sami Lepisto Hooking against Michael Grier|
|Jim Vandermeer Roughing against Patrick Kaleta|
|Paul Gaustad Slashing against Daniel Winnik|
|Scottie Upshall Roughing against Patrick Kaleta|
|Scottie Upshall Hi-sticking against Patrick Kaleta|
|Jim Vandermeer Cross checking against Thomas Vanek|